We will use separation of variables to derive the general solution for the exponential growth model.

                          Let 

                                      is just a constant so  will also just be some constant.  We let .

In the exponential growth model (in this case it would be called the exponential decay model), .  But the entire exponent can be negative;  causing an exponentially decreasing population until that population reaches zero.  In theory, it would continue into negative values but biologically we know this is not feasible.

Logistic growth model is very similar to the exponential growth model except now we are taking into account a carrying capacity.  At some point, a population will grow so large the surrounding resources can no longer support it.  Before it reaches that point, there is a stable equilibrium where the population can be supported by the resources available if it stays at a constant number of individuals.  The equilibrium is defined by the carrying capacity.

The exponential growth model is used to show how populations grow over time.  This model shows a population growing exponentially without a carrying capacity limiting the population at some point.   is the growth constant and  is the population.

The most simple exponential growth model only takes into account the population’s current state, , time, , and a growth/decay constant that is greater than zero .  It does not limit the population to a carrying capacity or take into account resource availability/ predator-prey interactions.

We will use separation of variables to solve this differential equation.

We use partial fractions for the left hand side:

It is clear that  so then  so our partial fraction decomposition:

Plugging back into our separation of variables:

                            Let 

                       Let 

We will evaluate at  in order to solve for .  Evaluating at  gives .  We will substitute this in for  into the equation we are solving.

Notice how the graph grows exponentially until it reaches a certain equilibrium.  This tells us that the population is approaching a carrying capacity so this graph shows logistic growth.  The other graph depicts exponential growth.

This example states a carrying capacity for the population.  Since exponential growth does not take into account carrying capacity, we cannot use this model for the population.  So we must use logistic growth.

The logistic growth model models a population taking into account a carrying capacity; that is, how large a population can grow and still survive off of available resources.  If the population is below this carrying capacity, it will grow to meet it.  If the population is higher than this carrying capacity, it will decrease to the carrying capacity.  This makes the carrying capacity a stable equilibrium point of the population.

This example only gives use a growth rate and a starting point for the population.  There is no limiting factor or carrying capacity so we must use exponential growth to model this population.